This talk is based on data from a study of the development of mathematical ideas and ways of reasoning of a focus group of students who did mathematics together throughout their public school and early university years. Findings from the study, now in its seventeenth year, indicate how students' early ideas and images are later elaborated and presented in symbolic expressions of generalized mathematical ideas. From interview data, we hear from participants as young adults how they viewed their mathematical activity in structuring their investigations and justifying their solutions. Video episodes from the study will be shown to illustrate student reasoning over the years. From interviews conducted in their senior-high school and early university years, we learn about particular conditions that were important to participants in doing meaningful mathematics. The Private Universe Project in Mathematics (PUP-Math), produced by the Science Media Group of the Harvard Astrophysical Observatory, illustrates with six, one-hour, videotapes the first twelve years of the study. The tapes and accompanying materials are available at their website, http://www.learner.org. Sample problems from the long-term study are available on the website for the Robert B. Davis Institute for Learning (RBDIL), Rutgers University: www.gse.rutgers.edu/rbdil.